Preimage problems for deterministic finite automata
Abstract
Given a subset of states of a deterministic finite automaton and a word , the preimage is the subset of all states mapped to a state in by the action of . We study three natural problems concerning words giving certain preimages. The first problem is whether, for a given subset, there exists a word \emph{extending} the subset (giving a larger preimage). The second problem is whether there exists a \emph{totally extending} word (giving the whole set of states as a preimage)---equivalently, whether there exists an \emph{avoiding} word for the complementary subset. The third problem is whether there exists a \emph{resizing} word. We also consider variants where the length of the word is upper bounded, where the size of the given subset is restricted, and where the automaton is strongly connected, synchronizing, or binary. We conclude with a summary of the complexities in all combinations of the cases.
Keywords
Cite
@article{arxiv.1704.08233,
title = {Preimage problems for deterministic finite automata},
author = {Mikhail V. Berlinkov and Robert Ferens and Marek Szykuła},
journal= {arXiv preprint arXiv:1704.08233},
year = {2020}
}